A133161 Indices of the triangular numbers which are also centered triangular number.

1, 4, 16, 61, 229, 856, 3196, 11929, 44521, 166156, 620104, 2314261, 8636941, 32233504, 120297076, 448954801, 1675522129, 6253133716, 23337012736, 87094917229, 325042656181, 1213075707496, 4527260173804, 16895964987721, 63056599777081, 235330434120604, 878265136705336

Offset

1,2

Comments

Also, indices of the triangular numbers which are sums of three consecutive triangular numbers (see A129803).

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 59.

Links

Table of n, a(n) for n=1..27.

A. Kozikowska, Topological classes of statically determinate beams with arbitrary number of supports, Journal of Theoretical and Applied Mechanics, 49, 4, pp. 1079-1100, Warsaw 2011; (see Eq. 5.18). - N. J. A. Sloane, Dec 17 2011.

Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Formula

a(n+2) = 4*a(n+1)-a(n)+1.

a(n+1) = 2*a(n) + 1/2 + (1/2)*(12*a(n)^2+12*a(n)-15)^(1/2).

G.f.: x*(1-x+x^2)/(1-x)/(1-4*x+x^2). - R. J. Mathar, Oct 24 2007

a(n)-a(n-1)= A005320(n-1). - R. J. Mathar, Mar 14 2016

E.g.f.: exp(x)*(exp(x)*(3*cosh(sqrt(3)*x) - sqrt(3)*sinh(sqrt(3)*x)) - 1)/2 - 1. - Stefano Spezia, Oct 18 2025

Mathematica

LinearRecurrence[{5, -5, 1}, {1, 4, 16}, 30] (* Harvey P. Dale, Aug 29 2017 *)

Crossrefs

Cf. A001834, A102871, A128862, A129803, A001571.

Sequence in context: A126929 A338531 A268452 * A103820 A206570 A206790

Adjacent sequences: A133158 A133159 A133160 * A133162 A133163 A133164

Keyword

nonn,easy

Author

Richard Choulet, Oct 09 2007

Extensions

a(25)-a(27) from Stefano Spezia, Oct 18 2025

Status

approved